Stability Condition for Circular Orbit

[cpburris]

Homework Statement

Show that the stability condition for a circular orbit of radius a, i.e.

[itex]f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0 [/itex]

is equivalent to the condition

[itex] \frac{d^2V(r)}{dr^2} > 0 [/itex]

for r=a where V(r) is the effective potential given by

[itex] V(r) = U(r) + \frac{ml^2}{2r^2} [/itex]

The Attempt at a Solution

I understand fully why they are equivalent, and I would have no problem proving individually how each is a condition for stability, but analytically I really don't know how to show the two are equivalent. I'm not even sure what the question is asking. I tried just setting

[itex] -\frac{d^2V(r)}{dr^2} = f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} [/itex]

and do something from there, but it didn't get me anywhere.

 

[BruceW]

I'm guessing that you already know what ##f(r)## is as a function of the potential? So if you use this definition, you could write out ##f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0## in terms of the potential instead, and start to see how it could be similar to the other equation.